Axiom of Dependent Choice

Let $S$ be a nonempty set, let $\mathbb{N}$ denote the natural numbers, and let $R$ be a binary relation on $S$, that is, a subset of the Cartesian product $S\times S$. Assume that for every $x\in S$ there exists $y\in S$ with $(x,y)\in R$. Let $s\in S$.

Then there exists a sequence $(a_m)_{m\in\mathbb{N}}$ in $S$ such that $a_1=s$ and $(a_m,a_{m+1})\in R$ for every $m\in\mathbb{N}$.